Hands-on Ex 2A

1st Order Spatial Point Pattern Analysis Methods

Author

Stephen Tay

Published

August 31, 2024

Modified

August 31, 2024

1. Overview

Point pattern analysis examines the spatial distribution of points and the underlying characteristics of these patterns. The properties of point patterns are typically categorised into two main types:

  • First-order properties: These focus on the distribution and variation of individual locations across space, assessing how the intensity of points varies over the study area.
  • Second-order properties: These consider the interactions between points, analyzing how points influence each other and how these interactions shape the overall pattern.

This exercise is centered on first-order properties. In general, density-based methods, such as kernel density estimation, primarily address first-order properties by examining the spatial intensity of points across the study area. In addition to the geospatial tools introduced in previous hands-on exercises, we will also utilize the spatstat, raster, and maptools packages.

pacman::p_load(sf, spatstat, raster, maptools, tmap, tidyverse)

2. Importing & Transforming Data

We will be importing 3 datasets:

  • Locations of childcare centers from data.gov.sg
  • Master Plan 2014 Subzone Boundary (Web) from data.gov.sg
  • National Coastal Boundary of Singapore from Singapore Land Authority (SLA)

Since the original datasets were in geographic coordinate system or had incorrect projection system, we will need to transform them to the same projection system.

childcare_sf <- st_read("data/geospatial/child-care-services-geojson.geojson") %>%
  st_transform(crs = 3414)
Reading layer `child-care-services-geojson' from data source 
  `/Users/stephentay/stephentay/ISSS626-Geospatial-Analytics/Hands-on_Ex/Hands-on_Ex02/data/geospatial/child-care-services-geojson.geojson' 
  using driver `GeoJSON'
Simple feature collection with 1545 features and 2 fields
Geometry type: POINT
Dimension:     XYZ
Bounding box:  xmin: 103.6824 ymin: 1.248403 xmax: 103.9897 ymax: 1.462134
z_range:       zmin: 0 zmax: 0
Geodetic CRS:  WGS 84
mpsz_sf <- st_read(dsn = "data/geospatial", layer = "MP14_SUBZONE_WEB_PL") %>%
  st_set_crs(3414)
Reading layer `MP14_SUBZONE_WEB_PL' from data source 
  `/Users/stephentay/stephentay/ISSS626-Geospatial-Analytics/Hands-on_Ex/Hands-on_Ex02/data/geospatial' 
  using driver `ESRI Shapefile'
Simple feature collection with 323 features and 15 fields
Geometry type: MULTIPOLYGON
Dimension:     XY
Bounding box:  xmin: 2667.538 ymin: 15748.72 xmax: 56396.44 ymax: 50256.33
Projected CRS: SVY21
Warning: st_crs<- : replacing crs does not reproject data; use st_transform for
that
st_crs(mpsz_sf)
Coordinate Reference System:
  User input: EPSG:3414 
  wkt:
PROJCRS["SVY21 / Singapore TM",
    BASEGEOGCRS["SVY21",
        DATUM["SVY21",
            ELLIPSOID["WGS 84",6378137,298.257223563,
                LENGTHUNIT["metre",1]]],
        PRIMEM["Greenwich",0,
            ANGLEUNIT["degree",0.0174532925199433]],
        ID["EPSG",4757]],
    CONVERSION["Singapore Transverse Mercator",
        METHOD["Transverse Mercator",
            ID["EPSG",9807]],
        PARAMETER["Latitude of natural origin",1.36666666666667,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8801]],
        PARAMETER["Longitude of natural origin",103.833333333333,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8802]],
        PARAMETER["Scale factor at natural origin",1,
            SCALEUNIT["unity",1],
            ID["EPSG",8805]],
        PARAMETER["False easting",28001.642,
            LENGTHUNIT["metre",1],
            ID["EPSG",8806]],
        PARAMETER["False northing",38744.572,
            LENGTHUNIT["metre",1],
            ID["EPSG",8807]]],
    CS[Cartesian,2],
        AXIS["northing (N)",north,
            ORDER[1],
            LENGTHUNIT["metre",1]],
        AXIS["easting (E)",east,
            ORDER[2],
            LENGTHUNIT["metre",1]],
    USAGE[
        SCOPE["Cadastre, engineering survey, topographic mapping."],
        AREA["Singapore - onshore and offshore."],
        BBOX[1.13,103.59,1.47,104.07]],
    ID["EPSG",3414]]
sg_sf <- st_read(dsn = "data/geospatial", layer="CostalOutline") %>%
  st_set_crs(3414)
Reading layer `CostalOutline' from data source 
  `/Users/stephentay/stephentay/ISSS626-Geospatial-Analytics/Hands-on_Ex/Hands-on_Ex02/data/geospatial' 
  using driver `ESRI Shapefile'
Simple feature collection with 60 features and 4 fields
Geometry type: POLYGON
Dimension:     XY
Bounding box:  xmin: 2663.926 ymin: 16357.98 xmax: 56047.79 ymax: 50244.03
Projected CRS: SVY21
Warning: st_crs<- : replacing crs does not reproject data; use st_transform for
that
st_crs(sg_sf)
Coordinate Reference System:
  User input: EPSG:3414 
  wkt:
PROJCRS["SVY21 / Singapore TM",
    BASEGEOGCRS["SVY21",
        DATUM["SVY21",
            ELLIPSOID["WGS 84",6378137,298.257223563,
                LENGTHUNIT["metre",1]]],
        PRIMEM["Greenwich",0,
            ANGLEUNIT["degree",0.0174532925199433]],
        ID["EPSG",4757]],
    CONVERSION["Singapore Transverse Mercator",
        METHOD["Transverse Mercator",
            ID["EPSG",9807]],
        PARAMETER["Latitude of natural origin",1.36666666666667,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8801]],
        PARAMETER["Longitude of natural origin",103.833333333333,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8802]],
        PARAMETER["Scale factor at natural origin",1,
            SCALEUNIT["unity",1],
            ID["EPSG",8805]],
        PARAMETER["False easting",28001.642,
            LENGTHUNIT["metre",1],
            ID["EPSG",8806]],
        PARAMETER["False northing",38744.572,
            LENGTHUNIT["metre",1],
            ID["EPSG",8807]]],
    CS[Cartesian,2],
        AXIS["northing (N)",north,
            ORDER[1],
            LENGTHUNIT["metre",1]],
        AXIS["easting (E)",east,
            ORDER[2],
            LENGTHUNIT["metre",1]],
    USAGE[
        SCOPE["Cadastre, engineering survey, topographic mapping."],
        AREA["Singapore - onshore and offshore."],
        BBOX[1.13,103.59,1.47,104.07]],
    ID["EPSG",3414]]

3. GeoVisualisation of Childcare Centers

It’s important to take a look at the spatial patterns of the childcare centers.

tm_shape(sg_sf) +
  tm_polygons() +
  tm_shape(mpsz_sf) +
  tm_polygons() +
  tm_shape(childcare_sf) +
  tm_dots(size = 0.01, alpha = 0.5)

An interactive pin map allows for free navigation and zooming across the map. You can easily query information about each point by clicking on it. Additionally, you have the option to change the background layer of the map.

tmap_mode('view')
tmap mode set to interactive viewing
tm_shape(childcare_sf)+
  tm_dots()
tmap_mode('plot')
tmap mode set to plotting

4. Geospatial Data Wrangling

Many geospatial analysis packages require input data to be in the Spatial class from the sp package.

For our analysis, we only need to convert the childcare dataset into ppp format for use with the spatstat package. However, we will also demonstrate how to convert simple feature dataframes to other formats for your reference.

childcare <- as_Spatial(childcare_sf)
mpsz <- as_Spatial(mpsz_sf)
sg <- as_Spatial(sg_sf)

class(childcare)
[1] "SpatialPointsDataFrame"
attr(,"package")
[1] "sp"
class(mpsz)
[1] "SpatialPolygonsDataFrame"
attr(,"package")
[1] "sp"
class(sg)
[1] "SpatialPolygonsDataFrame"
attr(,"package")
[1] "sp"
childcare
class       : SpatialPointsDataFrame 
features    : 1545 
extent      : 11203.01, 45404.24, 25667.6, 49300.88  (xmin, xmax, ymin, ymax)
crs         : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs 
variables   : 2
names       :    Name,                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           Description 
min values  :   kml_1, <center><table><tr><th colspan='2' align='center'><em>Attributes</em></th></tr><tr bgcolor="#E3E3F3"> <th>ADDRESSBLOCKHOUSENUMBER</th> <td></td> </tr><tr bgcolor=""> <th>ADDRESSBUILDINGNAME</th> <td></td> </tr><tr bgcolor="#E3E3F3"> <th>ADDRESSPOSTALCODE</th> <td>018989</td> </tr><tr bgcolor=""> <th>ADDRESSSTREETNAME</th> <td>1, MARINA BOULEVARD, #B1 - 01, ONE MARINA BOULEVARD, SINGAPORE 018989</td> </tr><tr bgcolor="#E3E3F3"> <th>ADDRESSTYPE</th> <td></td> </tr><tr bgcolor=""> <th>DESCRIPTION</th> <td></td> </tr><tr bgcolor="#E3E3F3"> <th>HYPERLINK</th> <td></td> </tr><tr bgcolor=""> <th>LANDXADDRESSPOINT</th> <td>0</td> </tr><tr bgcolor="#E3E3F3"> <th>LANDYADDRESSPOINT</th> <td>0</td> </tr><tr bgcolor=""> <th>NAME</th> <td>THE LITTLE SKOOL-HOUSE INTERNATIONAL PTE. LTD.</td> </tr><tr bgcolor="#E3E3F3"> <th>PHOTOURL</th> <td></td> </tr><tr bgcolor=""> <th>ADDRESSFLOORNUMBER</th> <td></td> </tr><tr bgcolor="#E3E3F3"> <th>INC_CRC</th> <td>08F73931F4A691F4</td> </tr><tr bgcolor=""> <th>FMEL_UPD_D</th> <td>20200826094036</td> </tr><tr bgcolor="#E3E3F3"> <th>ADDRESSUNITNUMBER</th> <td></td> </tr></table></center> 
max values  : kml_999,                  <center><table><tr><th colspan='2' align='center'><em>Attributes</em></th></tr><tr bgcolor="#E3E3F3"> <th>ADDRESSBLOCKHOUSENUMBER</th> <td></td> </tr><tr bgcolor=""> <th>ADDRESSBUILDINGNAME</th> <td></td> </tr><tr bgcolor="#E3E3F3"> <th>ADDRESSPOSTALCODE</th> <td>829646</td> </tr><tr bgcolor=""> <th>ADDRESSSTREETNAME</th> <td>200, PONGGOL SEVENTEENTH AVENUE, SINGAPORE 829646</td> </tr><tr bgcolor="#E3E3F3"> <th>ADDRESSTYPE</th> <td></td> </tr><tr bgcolor=""> <th>DESCRIPTION</th> <td>Child Care Services</td> </tr><tr bgcolor="#E3E3F3"> <th>HYPERLINK</th> <td></td> </tr><tr bgcolor=""> <th>LANDXADDRESSPOINT</th> <td>0</td> </tr><tr bgcolor="#E3E3F3"> <th>LANDYADDRESSPOINT</th> <td>0</td> </tr><tr bgcolor=""> <th>NAME</th> <td>RAFFLES KIDZ @ PUNGGOL PTE LTD</td> </tr><tr bgcolor="#E3E3F3"> <th>PHOTOURL</th> <td></td> </tr><tr bgcolor=""> <th>ADDRESSFLOORNUMBER</th> <td></td> </tr><tr bgcolor="#E3E3F3"> <th>INC_CRC</th> <td>379D017BF244B0FA</td> </tr><tr bgcolor=""> <th>FMEL_UPD_D</th> <td>20200826094036</td> </tr><tr bgcolor="#E3E3F3"> <th>ADDRESSUNITNUMBER</th> <td></td> </tr></table></center> 
mpsz
class       : SpatialPolygonsDataFrame 
features    : 323 
extent      : 2667.538, 56396.44, 15748.72, 50256.33  (xmin, xmax, ymin, ymax)
crs         : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs 
variables   : 15
names       : OBJECTID, SUBZONE_NO, SUBZONE_N, SUBZONE_C, CA_IND, PLN_AREA_N, PLN_AREA_C,       REGION_N, REGION_C,          INC_CRC, FMEL_UPD_D,     X_ADDR,     Y_ADDR,    SHAPE_Leng,    SHAPE_Area 
min values  :        1,          1, ADMIRALTY,    AMSZ01,      N, ANG MO KIO,         AM, CENTRAL REGION,       CR, 00F5E30B5C9B7AD8,      16409,  5092.8949,  19579.069, 871.554887798, 39437.9352703 
max values  :      323,         17,    YUNNAN,    YSSZ09,      Y,     YISHUN,         YS,    WEST REGION,       WR, FFCCF172717C2EAF,      16409, 50424.7923, 49552.7904, 68083.9364708,  69748298.792 
sg
class       : SpatialPolygonsDataFrame 
features    : 60 
extent      : 2663.926, 56047.79, 16357.98, 50244.03  (xmin, xmax, ymin, ymax)
crs         : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs 
variables   : 4
names       : GDO_GID, MSLINK, MAPID,              COSTAL_NAM 
min values  :       1,      1,     0,             ISLAND LINK 
max values  :      60,     67,     0, SINGAPORE - MAIN ISLAND 
childcare_sp <- as(childcare, "SpatialPoints")
sg_sp <- as(sg, "SpatialPolygons")
childcare_sp
class       : SpatialPoints 
features    : 1545 
extent      : 11203.01, 45404.24, 25667.6, 49300.88  (xmin, xmax, ymin, ymax)
crs         : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs 
sg_sp
class       : SpatialPolygons 
features    : 60 
extent      : 2663.926, 56047.79, 16357.98, 50244.03  (xmin, xmax, ymin, ymax)
crs         : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs 

Here, we convert the childcare dataset into ppp format. We will also plot it and show the summary statistics of the created ppp object.

childcare_ppp <- as.ppp(childcare_sf)
Warning in as.ppp.sf(childcare_sf): only first attribute column is used for
marks
childcare_ppp
Marked planar point pattern: 1545 points
marks are of storage type  'character'
window: rectangle = [11203.01, 45404.24] x [25667.6, 49300.88] units
plot(childcare_ppp)
Warning in default.charmap(ntypes, chars): Too many types to display every type
as a different character
Warning: Only 10 out of 1545 symbols are shown in the symbol map

summary(childcare_ppp)
Marked planar point pattern:  1545 points
Average intensity 1.91145e-06 points per square unit

Coordinates are given to 11 decimal places

marks are of type 'character'
Summary:
   Length     Class      Mode 
     1545 character character 

Window: rectangle = [11203.01, 45404.24] x [25667.6, 49300.88] units
                    (34200 x 23630 units)
Window area = 808287000 square units

4.4 Handling duplicated points

any(duplicated(childcare_ppp))
[1] FALSE
sum(multiplicity(childcare_ppp) > 1)
[1] 0
tmap_mode('view')
tmap mode set to interactive viewing
tm_shape(childcare) +
  tm_dots(alpha=0.4, 
          size=0.05)
tmap_mode('plot')
tmap mode set to plotting

The simplest way to handle duplicated points is to delete them, but this approach may result in the loss of valuable data. Alternatively, you can use jittering, which applies a small perturbation to the duplicate points, ensuring they do not occupy the exact same location.

childcare_ppp_jit <- rjitter(childcare_ppp, 
                             retry=TRUE, 
                             nsim=1, 
                             drop=TRUE)
any(duplicated(childcare_ppp_jit))
[1] FALSE
sum(multiplicity(childcare_ppp_jit) > 1)
[1] 0

4.5 Create owin object

When analyzing spatial point patterns, it is good practice to confine the analysis within a geographical boundary. This involves converting a spatial polygon into an owin object for use with the spatstat package later.

sg_owin <- as.owin(sg_sf)
plot(sg_owin)

summary(sg_owin)
Window: polygonal boundary
50 separate polygons (1 hole)
                 vertices         area relative.area
polygon 1 (hole)       30     -7081.18     -9.76e-06
polygon 2              55     82537.90      1.14e-04
polygon 3              90    415092.00      5.72e-04
polygon 4              49     16698.60      2.30e-05
polygon 5              38     24249.20      3.34e-05
polygon 6             976  23344700.00      3.22e-02
polygon 7             721   1927950.00      2.66e-03
polygon 8            1992   9992170.00      1.38e-02
polygon 9             330   1118960.00      1.54e-03
polygon 10            175    925904.00      1.28e-03
polygon 11            115    928394.00      1.28e-03
polygon 12             24      6352.39      8.76e-06
polygon 13            190    202489.00      2.79e-04
polygon 14             37     10170.50      1.40e-05
polygon 15             25     16622.70      2.29e-05
polygon 16             10      2145.07      2.96e-06
polygon 17             66     16184.10      2.23e-05
polygon 18           5195 636837000.00      8.78e-01
polygon 19             76    312332.00      4.31e-04
polygon 20            627  31891300.00      4.40e-02
polygon 21             20     32842.00      4.53e-05
polygon 22             42     55831.70      7.70e-05
polygon 23             67   1313540.00      1.81e-03
polygon 24            734   4690930.00      6.47e-03
polygon 25             16      3194.60      4.40e-06
polygon 26             15      4872.96      6.72e-06
polygon 27             15      4464.20      6.15e-06
polygon 28             14      5466.74      7.54e-06
polygon 29             37      5261.94      7.25e-06
polygon 30            111    662927.00      9.14e-04
polygon 31             69     56313.40      7.76e-05
polygon 32            143    145139.00      2.00e-04
polygon 33            397   2488210.00      3.43e-03
polygon 34             90    115991.00      1.60e-04
polygon 35             98     62682.90      8.64e-05
polygon 36            165    338736.00      4.67e-04
polygon 37            130     94046.50      1.30e-04
polygon 38             93    430642.00      5.94e-04
polygon 39             16      2010.46      2.77e-06
polygon 40            415   3253840.00      4.49e-03
polygon 41             30     10838.20      1.49e-05
polygon 42             53     34400.30      4.74e-05
polygon 43             26      8347.58      1.15e-05
polygon 44             74     58223.40      8.03e-05
polygon 45            327   2169210.00      2.99e-03
polygon 46            177    467446.00      6.44e-04
polygon 47             46    699702.00      9.65e-04
polygon 48              6     16841.00      2.32e-05
polygon 49             13     70087.30      9.66e-05
polygon 50              4      9459.63      1.30e-05
enclosing rectangle: [2663.93, 56047.79] x [16357.98, 50244.03] units
                     (53380 x 33890 units)
Window area = 725376000 square units
Fraction of frame area: 0.401

4.6 Combine owin object with point events

In this final step of geospatial data wrangling, we will extract childcare events located within Singapore boundary using the code chunk below. The resulting object combines both point and polygon features into a single ppp object.

childcareSG_ppp = childcare_ppp[sg_owin]
summary(childcareSG_ppp)
Marked planar point pattern:  1545 points
Average intensity 2.129929e-06 points per square unit

Coordinates are given to 11 decimal places

marks are of type 'character'
Summary:
   Length     Class      Mode 
     1545 character character 

Window: polygonal boundary
50 separate polygons (1 hole)
                 vertices         area relative.area
polygon 1 (hole)       30     -7081.18     -9.76e-06
polygon 2              55     82537.90      1.14e-04
polygon 3              90    415092.00      5.72e-04
polygon 4              49     16698.60      2.30e-05
polygon 5              38     24249.20      3.34e-05
polygon 6             976  23344700.00      3.22e-02
polygon 7             721   1927950.00      2.66e-03
polygon 8            1992   9992170.00      1.38e-02
polygon 9             330   1118960.00      1.54e-03
polygon 10            175    925904.00      1.28e-03
polygon 11            115    928394.00      1.28e-03
polygon 12             24      6352.39      8.76e-06
polygon 13            190    202489.00      2.79e-04
polygon 14             37     10170.50      1.40e-05
polygon 15             25     16622.70      2.29e-05
polygon 16             10      2145.07      2.96e-06
polygon 17             66     16184.10      2.23e-05
polygon 18           5195 636837000.00      8.78e-01
polygon 19             76    312332.00      4.31e-04
polygon 20            627  31891300.00      4.40e-02
polygon 21             20     32842.00      4.53e-05
polygon 22             42     55831.70      7.70e-05
polygon 23             67   1313540.00      1.81e-03
polygon 24            734   4690930.00      6.47e-03
polygon 25             16      3194.60      4.40e-06
polygon 26             15      4872.96      6.72e-06
polygon 27             15      4464.20      6.15e-06
polygon 28             14      5466.74      7.54e-06
polygon 29             37      5261.94      7.25e-06
polygon 30            111    662927.00      9.14e-04
polygon 31             69     56313.40      7.76e-05
polygon 32            143    145139.00      2.00e-04
polygon 33            397   2488210.00      3.43e-03
polygon 34             90    115991.00      1.60e-04
polygon 35             98     62682.90      8.64e-05
polygon 36            165    338736.00      4.67e-04
polygon 37            130     94046.50      1.30e-04
polygon 38             93    430642.00      5.94e-04
polygon 39             16      2010.46      2.77e-06
polygon 40            415   3253840.00      4.49e-03
polygon 41             30     10838.20      1.49e-05
polygon 42             53     34400.30      4.74e-05
polygon 43             26      8347.58      1.15e-05
polygon 44             74     58223.40      8.03e-05
polygon 45            327   2169210.00      2.99e-03
polygon 46            177    467446.00      6.44e-04
polygon 47             46    699702.00      9.65e-04
polygon 48              6     16841.00      2.32e-05
polygon 49             13     70087.30      9.66e-05
polygon 50              4      9459.63      1.30e-05
enclosing rectangle: [2663.93, 56047.79] x [16357.98, 50244.03] units
                     (53380 x 33890 units)
Window area = 725376000 square units
Fraction of frame area: 0.401
plot(childcareSG_ppp)
Warning in default.charmap(ntypes, chars): Too many types to display every type
as a different character
Warning: Only 10 out of 1545 symbols are shown in the symbol map

5. Kernel Density Estimation (KDE)

5.1 Introduction to KDE

The code chunk below computes kernel density using the following configurations in the density() function:

  • sigma: Specifies the automatic bandwidth selection method. Recommended methods are bw.diggle, bw.ppl, bw.CvL, or bw.scott.
  • kernel: Specifies the smoothing method to be used, e.g. “gaussian”, “epanechnikov”, “quartic” or “disc”.
  • edge: If set to TRUE, the intensity estimate is corrected for edge effect bias using the method described by Jones (1993) and Diggle (2010, equation 18.9).
kde_childcareSG_bw <- density(childcareSG_ppp,
                              sigma=bw.diggle,
                              kernel="gaussian",
                              edge=TRUE) 

This is how the KDE plot looks like. The density values in the output range from 0 to 0.00003, which is too small to easily interpret. This occurs because the default unit of measurement in the SVY21 coordinate system is meters. Consequently, the computed density values are in “number of points per square meter”.

plot(kde_childcareSG_bw)

To check the bandwidth used to compute the KDE layer, use the code chunk below.

bw.diggle(childcareSG_ppp)
   sigma 
298.4095 

5.2 Rescaling KDE values

rescale.ppp() is used to convert the unit of measurement from meter to kilometer.

childcareSG_ppp.km <- rescale.ppp(childcareSG_ppp, 1000, "km")

Notice that output image looks identical to the earlier version, the only changes in the data values (refer to the legend).

kde_childcareSG.bw <- density(childcareSG_ppp.km, sigma=bw.diggle, kernel="gaussian", edge=TRUE)
plot(kde_childcareSG.bw)

5.3 Working with different automatic bandwidth methods

Let’s take a look at the bandwidth of the various automatic bandwidth methods.

bw.diggle(childcareSG_ppp.km)
    sigma 
0.2984095 
bw.ppl(childcareSG_ppp.km)
    sigma 
0.3897114 
bw.CvL(childcareSG_ppp.km)
   sigma 
4.543278 
bw.scott(childcareSG_ppp.km)
 sigma.x  sigma.y 
2.224898 1.450966 

Here, we will compare the KDEs generated by various automatic bandwidth selection methods. Baddeley et al. (2016) recommend using the bw.ppl algorithm, as their experience shows it tends to produce more appropriate values when the point pattern consists predominantly of tight clusters.

kde_childcareSG.ppl <- density(childcareSG_ppp.km, sigma=bw.ppl, kernel="gaussian", edge=TRUE)
kde_childcareSG.cvl <- density(childcareSG_ppp.km, sigma=bw.CvL, kernel="gaussian", edge=TRUE)
kde_childcareSG.sct <- density(childcareSG_ppp.km, sigma=bw.scott, kernel="gaussian", edge=TRUE)

par(mfrow=c(2,2))
plot(kde_childcareSG.bw, main = "bw.diggle")
plot(kde_childcareSG.ppl, main = "bw.ppl")
plot(kde_childcareSG.cvl, main = "bw.CvL")
plot(kde_childcareSG.sct, main = "bw.scott")

5.4 Working with different kernel methods

By default, the kernel method used in density.ppp() is Gaussian. However, there are three other options available: Epanechnikov, Quartic, and Disc.

par(mfrow=c(2,2))
plot(density(childcareSG_ppp.km, 
             sigma=bw.ppl, 
             edge=TRUE, 
             kernel="gaussian"),
     main="Gaussian")
plot(density(childcareSG_ppp.km, 
             sigma=bw.ppl, 
             edge=TRUE, 
             kernel="epanechnikov"), 
     main="Epanechnikov")
Warning in density.ppp(childcareSG_ppp.km, sigma = bw.ppl, edge = TRUE, :
Bandwidth selection will be based on Gaussian kernel
plot(density(childcareSG_ppp.km, 
             sigma=bw.ppl, 
             edge=TRUE, 
             kernel="quartic"), 
     main="Quartic")
Warning in density.ppp(childcareSG_ppp.km, sigma = bw.ppl, edge = TRUE, :
Bandwidth selection will be based on Gaussian kernel
plot(density(childcareSG_ppp.km, 
             sigma=bw.ppl, 
             edge=TRUE, 
             kernel="disc"), 
     main="Disc")
Warning in density.ppp(childcareSG_ppp.km, sigma = bw.ppl, edge = TRUE, :
Bandwidth selection will be based on Gaussian kernel

5.5 Fixed and Adaptive KDE

You could also compute KDE using fixed or adaptive bandwidths.

Example: defining a bandwidth of 600 meter. Sigma value is 0.6, as the unit of measurement is kilometers.

kde_childcareSG_600 <- density(childcareSG_ppp.km, sigma=0.6, edge=TRUE, kernel="gaussian")
plot(kde_childcareSG_600)

The fixed bandwidth method is highly sensitive to skewed distributions of spatial point patterns, such as those found in urban versus rural areas. To address this issue, an adaptive bandwidth method can be used instead.

kde_childcareSG_adaptive <- adaptive.density(childcareSG_ppp.km, method="kernel")
plot(kde_childcareSG_adaptive)

par(mfrow=c(1,2))
plot(kde_childcareSG.bw, main = "Fixed bandwidth")
plot(kde_childcareSG_adaptive, main = "Adaptive bandwidth")

5.6 Convert KDE Output into Grid Object

We convert the KDE output so that it is suitable for mapping purposes.

gridded_kde_childcareSG_bw <- as.SpatialGridDataFrame.im(kde_childcareSG.bw)
spplot(gridded_kde_childcareSG_bw)

Notice that the crs property is NA.

kde_childcareSG_bw_raster <- raster(kde_childcareSG.bw)
kde_childcareSG_bw_raster
class      : RasterLayer 
dimensions : 128, 128, 16384  (nrow, ncol, ncell)
resolution : 0.4170614, 0.2647348  (x, y)
extent     : 2.663926, 56.04779, 16.35798, 50.24403  (xmin, xmax, ymin, ymax)
crs        : NA 
source     : memory
names      : layer 
values     : -1.005814e-14, 28.51831  (min, max)

The code chunk includes the CRS information onto the raster. Notice that the crs property is completed.

projection(kde_childcareSG_bw_raster) <- CRS("+init=EPSG:3414")
kde_childcareSG_bw_raster
class      : RasterLayer 
dimensions : 128, 128, 16384  (nrow, ncol, ncell)
resolution : 0.4170614, 0.2647348  (x, y)
extent     : 2.663926, 56.04779, 16.35798, 50.24403  (xmin, xmax, ymin, ymax)
crs        : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +units=m +no_defs 
source     : memory
names      : layer 
values     : -1.005814e-14, 28.51831  (min, max)

Finally, we will display the raster in cartographic quality map using tmap package.

tm_shape(kde_childcareSG_bw_raster) + 
  tm_raster("layer", palette = "viridis") +
  tm_layout(legend.position = c("right", "bottom"), frame = FALSE)

5.7 Comparing KDEs by Planning Areas

In this section, we will compare KDE of childcare locations at Punggol, Tampines, Choa Chu Kang and Jurong West planning areas.

pg <- mpsz_sf %>%
  filter(PLN_AREA_N == "PUNGGOL")
tm <- mpsz_sf %>%
  filter(PLN_AREA_N == "TAMPINES")
ck <- mpsz_sf %>%
  filter(PLN_AREA_N == "CHOA CHU KANG")
jw <- mpsz_sf %>%
  filter(PLN_AREA_N == "JURONG WEST")
plot(pg, main = "Punggol")
Warning: plotting the first 9 out of 15 attributes; use max.plot = 15 to plot
all

plot(tm, main = "Tampines")
Warning: plotting the first 9 out of 15 attributes; use max.plot = 15 to plot
all

plot(ck, main = "Choa Chu Kang")
Warning: plotting the first 10 out of 15 attributes; use max.plot = 15 to plot
all

plot(jw, main = "Jurong West")
Warning: plotting the first 9 out of 15 attributes; use max.plot = 15 to plot
all

pg_owin = as.owin(pg)
tm_owin = as.owin(tm)
ck_owin = as.owin(ck)
jw_owin = as.owin(jw)
childcare_pg_ppp = childcare_ppp_jit[pg_owin]
childcare_tm_ppp = childcare_ppp_jit[tm_owin]
childcare_ck_ppp = childcare_ppp_jit[ck_owin]
childcare_jw_ppp = childcare_ppp_jit[jw_owin]
childcare_pg_ppp.km = rescale.ppp(childcare_pg_ppp, 1000, "km")
childcare_tm_ppp.km = rescale.ppp(childcare_tm_ppp, 1000, "km")
childcare_ck_ppp.km = rescale.ppp(childcare_ck_ppp, 1000, "km")
childcare_jw_ppp.km = rescale.ppp(childcare_jw_ppp, 1000, "km")
par(mfrow=c(2,2))
plot(childcare_pg_ppp.km, main="Punggol")
Warning in default.charmap(ntypes, chars): Too many types to display every type
as a different character
Warning: Only 10 out of 61 symbols are shown in the symbol map
plot(childcare_tm_ppp.km, main="Tampines")
Warning in default.charmap(ntypes, chars): Too many types to display every type
as a different character
Warning: Only 10 out of 89 symbols are shown in the symbol map
plot(childcare_ck_ppp.km, main="Choa Chu Kang")
Warning in default.charmap(ntypes, chars): Too many types to display every type
as a different character
Warning: Only 10 out of 61 symbols are shown in the symbol map
plot(childcare_jw_ppp.km, main="Jurong West")
Warning in default.charmap(ntypes, chars): Too many types to display every type
as a different character
Warning: Only 10 out of 88 symbols are shown in the symbol map

par(mfrow=c(2,2))
plot(density(childcare_pg_ppp.km, 
             sigma=bw.diggle, 
             edge=TRUE, 
             kernel="gaussian"),
     main="Punggol")
Warning: Berman-Diggle Cross-Validation criterion was minimised at right-hand
end of interval [0, 0.245]; use argument 'hmax' to specify a wider interval for
bandwidth 'sigma'
plot(density(childcare_tm_ppp.km, 
             sigma=bw.diggle, 
             edge=TRUE, 
             kernel="gaussian"),
     main="Tampines")
plot(density(childcare_ck_ppp.km, 
             sigma=bw.diggle, 
             edge=TRUE, 
             kernel="gaussian"),
     main="Choa Chu Kang")
plot(density(childcare_jw_ppp.km, 
             sigma=bw.diggle, 
             edge=TRUE, 
             kernel="gaussian"),
     main="Jurong West")
Warning: Berman-Diggle Cross-Validation criterion was minimised at right-hand
end of interval [0, 0.281]; use argument 'hmax' to specify a wider interval for
bandwidth 'sigma'

For comparison purposes, we use 250m as the bandwidth.

par(mfrow=c(2,2))
plot(density(childcare_pg_ppp.km, 
             sigma=0.25, 
             edge=TRUE, 
             kernel="gaussian"),
     main="Punggol")
plot(density(childcare_tm_ppp.km, 
             sigma=0.25, 
             edge=TRUE, 
             kernel="gaussian"),
     main="Tampines")
plot(density(childcare_ck_ppp.km, 
             sigma=0.25, 
             edge=TRUE, 
             kernel="gaussian"),
     main="Choa Chu Kang")
plot(density(childcare_jw_ppp.km, 
             sigma=0.25, 
             edge=TRUE, 
             kernel="gaussian"),
     main="Jurong West")

6. Nearest Neighbor Analysis

In this section, we will conduct the Clark-Evans test for aggregation of a spatial point pattern using the clarkevans.test() function from spatstat.

The test hypotheses are as follows:

  • H0: The distribution of childcare locations is random.
  • H1: The distribution of childcare locations is not random.

A 95% confidence interval will be used.

Are childcare centers in Singapore clustered? The p-value is less than the alpha value and there is sufficient evidence to reject the null hypothesis, suggesting that the childcare centers in Singapore are clustered.

clarkevans.test(childcareSG_ppp,
                correction="none",
                clipregion="sg_owin",
                alternative=c("clustered"),
                nsim=99)

    Clark-Evans test
    No edge correction
    Z-test

data:  childcareSG_ppp
R = 0.55631, p-value < 2.2e-16
alternative hypothesis: clustered (R < 1)

Are childcare centers in CCK having clustered/regular pattern? The p-value is greater than the alpha value and there is insufficient evidence to reject the null hypothesis, suggesting that the childcare centers in CCK could be CSR.

clarkevans.test(childcare_ck_ppp,
                correction="none",
                clipregion=NULL,
                alternative=c("two.sided"),
                nsim=999)

    Clark-Evans test
    No edge correction
    Z-test

data:  childcare_ck_ppp
R = 0.9722, p-value = 0.6779
alternative hypothesis: two-sided

Are childcare centers in Tampines having clustered/regular pattern? The p-value is less than the alpha value and there is sufficient evidence to reject the null hypothesis, suggesting that the childcare centers in Tampines could be having clustered/regular pattern.

clarkevans.test(childcare_tm_ppp,
                correction="none",
                clipregion=NULL,
                alternative=c("two.sided"),
                nsim=999)

    Clark-Evans test
    No edge correction
    Z-test

data:  childcare_tm_ppp
R = 0.78556, p-value = 0.0001087
alternative hypothesis: two-sided